Data-driven, psuedo-spectral closures of the moment hierarchy for Vlasov-Poisson turbulence using machine learning*

A common approach to studying turbulence in magnetic fusion plasmas is to simulate microscale dynamics along magnetic field lines using gyrokinetic codes.  Several of these codes, such as GENE, CGYRO, and GX, employ pseudo-spectral methods, combining the accuracy of spectral methods with efficiency gained by avoiding convolutions in nonlinear terms in the spectral domain.  One potential avenue to accelerate these codes would be to increase their accuracy when operated at coarse resolution in phase space. To that end, we are exploring the potential to integrate machine-learning models of small-scale dynamics into coarse-resolution simulations.  We implemented a pseudo-spectral Eulerian code to solve the one-dimensional Vlasov-Poisson system on a basis of Fourier modes in configuration space and Hermite polynomials in velocity space.  When cast onto the Hermite basis, the Vlasov equation becomes an infinitely coupled hierarchy of fluid moments, presenting a closure problem.  In the linear limit of the system, we developed a machine-learning closure using a reservoir-computing architecture, leveraging its temporal memory.  When the kinetic Fourier-Hermite code is augmented with the reservoir closure, we report that the closure permits a reduction of the velocity resolution by a factor of ten, with a relative error within two percent for the moment at which the reservoir closes the hierarchy.  In our initial nonlinear studies, we report errors within ten percent for the moments containing the most energy.